But Krueger has refuted reports that Rodriguez, who is currently sidelined with a serious knee injury, and Schneiderlin will follow them away from the south coast.

"They are part of the core that we have decided to keep at Southampton," he told Sky Sports News. "Jay Rodriguez and Morgan Schneiderlin are not for sale and they will be a part of our club as we enter the new season.

"In and around Jay Rodriguez and Morgan Schneiderlin and the rest of our core players, we're excited to continue to build forward, but all those stories are false."

With the exception of Shaw, all of the players who have left the club have moved to sides which will be competing in the Champions League next season.

Jay Rodriguez has been linked with a transfer to Tottenham.

With a place in Europe's elite club competition the ultimate goal for Saints, who were playing in League One four seasons ago, Krueger accused those who have departed of taking the easy route to Champions League football.

"We entered the transfer window with a plan. We have a clear strategic approach to this summer and we did not put any players up for sale," he said. "It was quite clear that we had multiple players interested in taking a shortcut to Champions League football.

"We set premiums on the players and we reached those premiums. It was in the best interests of the club and in the best interests of our rebuild to make these deals and to continue to build forward with players and with the team that wants to go the hard way into Europe.

"There was interest there and we saw that interest brewing a little bit before the World Cup and quite clearly we stayed patient, we didn't make quick deals in any way, shape or form.

"None of these deals were made on financial reasons alone, we were completely ready to have players stay in Southampton even against their desires if we didn't reach our prices."

But Schneiderlin appeared to react badly to the comments, tweeting shortly afterwards:

New manager Ronald Koeman has already started the rebuilding job by bringing in Eredivisie pair Dusan Tadic and Graziano Pelle but the Dutchman will be keen to add several other players to his squad before the Premier League campaign begins at Liverpool.

Although Krueger knows adding more talent to the squad a must, he refused to put a time-frame on any new arrivals.

"We are excited to continue in the rebuilding process in the next few weeks," Krueger added. "We will have players coming in, we will make transfers to strengthen the squad. I can understand some frustration but we needed to stay silent to do the best deals for the club.

"Ronald Koeman has done an outstanding job so far. We are extremely pleased with the work going on at St Mary's and we know we are going in the right way but we also know that we need more players and that we need more depth.

"It is not about getting them in today or tomorrow, it is about getting the right players in on the right deal that fits the way Ronald wants to play. We want to give him the players that are necessary -- of course we would like to see them in sooner rather than later but we will make the deals when the time is right for the club.

"It is come to September that is judgement day and I think people need to realise we are in a process right now. The match is far from over and on Sept. 1 let's have a conversation."